# American Institute of Mathematical Sciences

January  2017, 22(1): 59-81. doi: 10.3934/dcdsb.2017003

## Stochastic volatility with regime switching and uncertain noise: Filtering with sub-linear expectations

 1 School of Mathematical Sciences, University of Adelaide, Adelaide, South Australia, Australia 2 Haskayne School of Business, University of Calgary, Calgary, Alberta, Canada 3 Centre for Applied Financial Studies, University of South Australia, Adelaide, South Australia, Australia 4 Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, NSW 2109, Australia

* Corresponding author: Robert J. Elliott

Received  January 2016 Revised  June 2016 Published  December 2016

Fund Project: This paper is dedicated to Professor K.L. Teo for his 70$^{th}$ birthday.

This paper considers a new stochastic volatility model with regime switches and uncertain noise in discrete time and discusses its theoretical development for filtering and estimation. The model incorporates important features for asset price models, such as stochastic volatility, regime switches and parameter uncertainty in Gaussian noises for both the return and volatility processes. In particular, both drift and volatility uncertainties for the return and volatility processes are incorporated by introducing a family of real-world probability measures. Then, by modifying the reference probability approach to filtering, a sequence of conditional sub-linear expectations is used to provide a robust approach for describing the drift and volatility uncertainties in the Gaussian noises. Filtering theory, based on conditional sublinear expectations and the Viterbi algorithm are adopted to derive filters for the hidden Markov chain and filter-based estimates of the unknown parameters.

Citation: Robert J. Elliott, Tak Kuen Siu. Stochastic volatility with regime switching and uncertain noise: Filtering with sub-linear expectations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 59-81. doi: 10.3934/dcdsb.2017003
##### References:
 [1] T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 31 (1986), 307-327.  doi: 10.1016/0304-4076(86)90063-1. [2] P. K. Clark, A subordinated stochastic process model with finite variance for speculative prices, Econometrica, 41 (1973), 135-155.  doi: 10.2307/1913889. [3] R. J. Elliott, L. Aggoun and J. Moore, Hidden Markov Models: Estimation and Control, $1^{st}$ edition, Springer-Verlag, New York, 1995. [4] R. J. Elliott, W. P. Malcolm and A. H. Tsoi, Robust parameter estimation for asset price models with Markov modulated volatilities, Journal of Economic Dynamics and Control, 27 (2003), 1391-1409.  doi: 10.1016/S0165-1889(02)00064-7. [5] R.J. Elliott and H. Miao, Stochastic volatility model with filtering, Stochastic Analysis and Applications, 24 (2006), 661-683.  doi: 10.1080/07362990600629389. [6] R. J. Elliott, J. van der Hoek and J. Valencia, Nonlinear filter estimation of volatility, Stochastic Analysis and Applications, 28 (2010), 696-710.  doi: 10.1080/07362994.2010.482841. [7] R. J. Elliott, C. C. Liew and T. K. Siu, On filtering and estimation of a threshold stochastic volatility model, Applied Mathematics and Computation, 218 (2011), 61-75.  doi: 10.1016/j.amc.2011.05.052. [8] R. J. Elliott, T. K. Siu and E. S. Fung, Filtering a nonlinear stochastic volatility model, Nonlinear Dynamics, 67 (2012), 1295-1313.  doi: 10.1007/s11071-011-0069-4. [9] R. J. Elliott, J. W. Lau, H. Miao and T. K. Siu, A Viterbi-based estimation for Markov switching GARCH model, Applied Mathematical Finance, 19 (2012), 219-231.  doi: 10.1080/1350486X.2011.620396. [10] R. J. Elliott, Filtering with uncertain noise, IEEE Transactions in Automatic Control, pp (2016), p1. doi: 10.1109/TAC. 2016. 2586585. [11] R. F. Engle, Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. Inflation, Econometrica, 50 (1982), 987-1007.  doi: 10.2307/1912773. [12] E. Ghysels, A. C. Harvey and E. Renault, Stochastic volatility, in Statistical Methods in Finance (eds. C. R. Rao and G. S. Maddala), North-Holland, 14 (1996), 119-191. doi: 10.1016/S0169-7161(96)14007-4. [13] J. D. Hamilton, A new approach to economic analysis of nonstationary time series and the business cycle, Econometrica, 57 (1989), 357-384.  doi: 10.2307/1912559. [14] L. P. Hansen and T. J. Sargent, Robustness, 1$^{st}$ edition, Princeton University Press, Princeton, 2008.  doi: 10.1515/9781400829385. [15] S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327. [16] J. C. Hull and A. White, The pricing of options on assets with stochastic volatilities, The Journal of Finance, 42 (1987), 281-300.  doi: 10.1111/j.1540-6261.1987.tb02568.x. [17] E. Jacquier, N. G. Polson and P. E. Rossi, Bayesian analysis of stochastic volatility models (with discussion), Journal of Business and Economics Statistics, 12 (1994), 371-417. [18] S. Kim, N. Shephard and S. Chib, Stochastic volatility: likelihood inference and comparison with ARCH models, Review of Economic Studies, 65 (1998), 361-393.  doi: 10.1111/1467-937X.00050. [19] S. Peng, $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô's type, in The Abel Symposium 2005, Abel Symposia 2 (eds. Benth et al. ), Springer-Verlag, 2 (2007), 541-567. doi: 10.1007/978-3-540-70847-6_25. [20] S. Peng, Nonlinear expectations and stochastic calculus under uncertainty, preprint, 1002. 4546. [21] M. K. Pitt and N. Shephard, Filtering via simulation: auxiliary particle filters, Journal of the American Statistical Association, 94 (1999), 590-599.  doi: 10.1080/01621459.1999.10474153. [22] N. Shephard, Stochastic Volatility: Selected Reading, $1^{st}$ edition, Oxford University Press, Oxford, 2005. [23] L. Scott, Option pricing when the variance changes randomly: Theory, estimation, and an application, Journal of Financial and Quantitative Analysis, 22 (1987), 419-438.  doi: 10.2307/2330793. [24] M. K. P. So, K. Lam and W. K. Li, A stochastic volatility model with Markov switching, Journal of Business and Economics Statistics, 16 (1998), 244-253.  doi: 10.2307/1392580. [25] E. M. Stein and J. C. Stein, Stock price distributions with stochastic volatility: An analytic approach, Review of Financial Studies, 4 (1991), 727-752.  doi: 10.1093/rfs/4.4.727. [26] G. E. Tauchen and M. Pitts, The price variability-volume relationship on speculative markets, Econometrica, 51 (1983), 485-505.  doi: 10.2307/1912002. [27] G. E. Tauchen, Stochastic volatility in general equilibrium, Quarterly Journal of Finance, 0 (2011), p707, http://dx.doi.org/10.1142/S2010139211000237 [28] S. J. Taylor, Financial returns modelled by the product of two stochastic processes, a study of daily sugar prices, 1961-79. in Time Series Analysis : Theory and Practice 1 (eds. O. D. Anderson), North Holland, (1982), 203-226. [29] S. J. Taylor, Modeling Financial Time Series, $1^{st}$ edition, Wiley, Chichester, 1986. [30] S. J. Taylor, Modeling stochastic volatility: A review and comparative study, Mathematical Finance, 4 (1994), 183-204.  doi: 10.1111/j.1467-9965.1994.tb00057.x. [31] S. J. Taylor, Asset Price Dynamics, Volatility and Prediction, $1^{st}$ edition, Princeton, Princeton University Press, 2005.  doi: 10.1515/9781400839254. [32] J. B. Wiggins, Option values under stochastic volatilities, Journal of Financial Economics, 19 (1987), 351-372.

show all references

##### References:
 [1] T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 31 (1986), 307-327.  doi: 10.1016/0304-4076(86)90063-1. [2] P. K. Clark, A subordinated stochastic process model with finite variance for speculative prices, Econometrica, 41 (1973), 135-155.  doi: 10.2307/1913889. [3] R. J. Elliott, L. Aggoun and J. Moore, Hidden Markov Models: Estimation and Control, $1^{st}$ edition, Springer-Verlag, New York, 1995. [4] R. J. Elliott, W. P. Malcolm and A. H. Tsoi, Robust parameter estimation for asset price models with Markov modulated volatilities, Journal of Economic Dynamics and Control, 27 (2003), 1391-1409.  doi: 10.1016/S0165-1889(02)00064-7. [5] R.J. Elliott and H. Miao, Stochastic volatility model with filtering, Stochastic Analysis and Applications, 24 (2006), 661-683.  doi: 10.1080/07362990600629389. [6] R. J. Elliott, J. van der Hoek and J. Valencia, Nonlinear filter estimation of volatility, Stochastic Analysis and Applications, 28 (2010), 696-710.  doi: 10.1080/07362994.2010.482841. [7] R. J. Elliott, C. C. Liew and T. K. Siu, On filtering and estimation of a threshold stochastic volatility model, Applied Mathematics and Computation, 218 (2011), 61-75.  doi: 10.1016/j.amc.2011.05.052. [8] R. J. Elliott, T. K. Siu and E. S. Fung, Filtering a nonlinear stochastic volatility model, Nonlinear Dynamics, 67 (2012), 1295-1313.  doi: 10.1007/s11071-011-0069-4. [9] R. J. Elliott, J. W. Lau, H. Miao and T. K. Siu, A Viterbi-based estimation for Markov switching GARCH model, Applied Mathematical Finance, 19 (2012), 219-231.  doi: 10.1080/1350486X.2011.620396. [10] R. J. Elliott, Filtering with uncertain noise, IEEE Transactions in Automatic Control, pp (2016), p1. doi: 10.1109/TAC. 2016. 2586585. [11] R. F. Engle, Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. Inflation, Econometrica, 50 (1982), 987-1007.  doi: 10.2307/1912773. [12] E. Ghysels, A. C. Harvey and E. Renault, Stochastic volatility, in Statistical Methods in Finance (eds. C. R. Rao and G. S. Maddala), North-Holland, 14 (1996), 119-191. doi: 10.1016/S0169-7161(96)14007-4. [13] J. D. Hamilton, A new approach to economic analysis of nonstationary time series and the business cycle, Econometrica, 57 (1989), 357-384.  doi: 10.2307/1912559. [14] L. P. Hansen and T. J. Sargent, Robustness, 1$^{st}$ edition, Princeton University Press, Princeton, 2008.  doi: 10.1515/9781400829385. [15] S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327. [16] J. C. Hull and A. White, The pricing of options on assets with stochastic volatilities, The Journal of Finance, 42 (1987), 281-300.  doi: 10.1111/j.1540-6261.1987.tb02568.x. [17] E. Jacquier, N. G. Polson and P. E. Rossi, Bayesian analysis of stochastic volatility models (with discussion), Journal of Business and Economics Statistics, 12 (1994), 371-417. [18] S. Kim, N. Shephard and S. Chib, Stochastic volatility: likelihood inference and comparison with ARCH models, Review of Economic Studies, 65 (1998), 361-393.  doi: 10.1111/1467-937X.00050. [19] S. Peng, $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô's type, in The Abel Symposium 2005, Abel Symposia 2 (eds. Benth et al. ), Springer-Verlag, 2 (2007), 541-567. doi: 10.1007/978-3-540-70847-6_25. [20] S. Peng, Nonlinear expectations and stochastic calculus under uncertainty, preprint, 1002. 4546. [21] M. K. Pitt and N. Shephard, Filtering via simulation: auxiliary particle filters, Journal of the American Statistical Association, 94 (1999), 590-599.  doi: 10.1080/01621459.1999.10474153. [22] N. Shephard, Stochastic Volatility: Selected Reading, $1^{st}$ edition, Oxford University Press, Oxford, 2005. [23] L. Scott, Option pricing when the variance changes randomly: Theory, estimation, and an application, Journal of Financial and Quantitative Analysis, 22 (1987), 419-438.  doi: 10.2307/2330793. [24] M. K. P. So, K. Lam and W. K. Li, A stochastic volatility model with Markov switching, Journal of Business and Economics Statistics, 16 (1998), 244-253.  doi: 10.2307/1392580. [25] E. M. Stein and J. C. Stein, Stock price distributions with stochastic volatility: An analytic approach, Review of Financial Studies, 4 (1991), 727-752.  doi: 10.1093/rfs/4.4.727. [26] G. E. Tauchen and M. Pitts, The price variability-volume relationship on speculative markets, Econometrica, 51 (1983), 485-505.  doi: 10.2307/1912002. [27] G. E. Tauchen, Stochastic volatility in general equilibrium, Quarterly Journal of Finance, 0 (2011), p707, http://dx.doi.org/10.1142/S2010139211000237 [28] S. J. Taylor, Financial returns modelled by the product of two stochastic processes, a study of daily sugar prices, 1961-79. in Time Series Analysis : Theory and Practice 1 (eds. O. D. Anderson), North Holland, (1982), 203-226. [29] S. J. Taylor, Modeling Financial Time Series, $1^{st}$ edition, Wiley, Chichester, 1986. [30] S. J. Taylor, Modeling stochastic volatility: A review and comparative study, Mathematical Finance, 4 (1994), 183-204.  doi: 10.1111/j.1467-9965.1994.tb00057.x. [31] S. J. Taylor, Asset Price Dynamics, Volatility and Prediction, $1^{st}$ edition, Princeton, Princeton University Press, 2005.  doi: 10.1515/9781400839254. [32] J. B. Wiggins, Option values under stochastic volatilities, Journal of Financial Economics, 19 (1987), 351-372.
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